Horizontal tangent planes to a surface

I already did this problem but there's something that bugs me. Here's how I solved it:

The normal vector to a curve at a point is
If this curve is to be horizontal, and must be zero, and the only component is in the z direction.

So, and this simplifies to

Similarly, by finding I get

So now I get two equatons:
(1)
(2)

By setting y=0 in (1), x=0 in (2).
I also get four other points

I did a few calculations and found 3 (!) tangent planes: including one at (0,0) which doesn't make sense... I even plotted the surface on Maple and saw that there cannot be a tangent plane at (0,0) because it would intersect the curve at other points.

And THAT is what I don't get... How come I get (0,0) as a point that has a horizontal tangent plane from my equations? Or should I just disregard this point when solving the problem for the same reason that I stated?
And how would I know for sure if I didn't have Maple, for example?